OPTICAL PUMPING OF SINGLE DONOR-BOUND ELECTRONS …rm270nf6941/thesis_sleiter_v4-augmented.pdfThis...

OPTICAL PUMPING OF SINGLE DONOR-BOUND ELECTRONS

IN ZINC SELENIDE AND SILICON

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Darin Jay Sleiter

August 2012

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This dissertation is online at: http://purl.stanford.edu/rm270nf6941

© 2012 by Darin Jay Sleiter. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii

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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Yoshihisa Yamamoto, Primary Adviser


David Goldhaber-Gordon


Jelena Vuckovic

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

The spin of single electrons bound to donors in semiconductor materials are promising

candidates for quantum bits implementations. These electrons have been shown to

be very homogenous and have extremely long decoherence times in a bulk semicon-

ductor environment. In this work, I have studied two donor systems as quantum bit

candidates, with a focus on using optical pumping methods to initialize and measure

the electron and nuclear spins of the donor system.

The first donor system, fluorine in zinc selenide, is a very optically bright sys-

tem which is a particularly good candidate for quantum repeater technologies. The

relatively large electron binding energy leads to a stable qubit at low temperatures,

and the potential for isotopic depletion of nuclear spin from the host semiconductor

crystal suggests very long decoherence times can be achieved. In this work, I confirm

the isolation of a single F-bound electron, and present results on the use of resonant

optical pumping to initialize the electron to a particular spin state. These results

open the door for optical control of the electron spin as a qubit.

The second donor system, phosphorus in silicon, is the semiconductor system with

the longest published decoherence times, obtained for the nuclear spin of the donor.

Due to the long excited-state lifetime of the donor optical transitions, the linewidth

of the transition is narrower than the hyperfine splitting, allowing optical access to

the donor nuclear spin. However, to date, single phosphorus donors have not been

optically isolated. In this work, I present a theoretical description of a hybrid optical

and electrical device for the measurement of a single phosphorus donor nuclear spin.

If experiments can confirm the properties of this device, this measurement technique

would provide a key element for a silicon-based quantum computer.

v

Acknowledgements

I have been extremely fortunate to have had a great deal of help and support from

numerous people throughout my time at Stanford, and this work certainly would not

have been possible without them.

I would first like to thank my advisor, Yoshihisa Yamamoto, for the opportunity he

gave me to work in his group on what I find to be such a fascinating area of research.

The freedom he has given me to work on a variety of topics, and to plan and execute

research on my own, has helped me develop my research skills, while his support

and insight into complex quantum behavior has kept the projects on a successful

path. I would also like to thank the other members of my reading committee: Jelena

Vuckovic and David Goldhaber-Gordon, both of whose research has been beneficial

to my understanding of the physics involved in the systems I’ve studied.

During the first few years of my time at Stanford, Thaddeus Ladd was a great

mentor to me, and I am very thankful for the opportunity to have worked with him.

His understanding of physics, ability to explain it, skills as an experimentalist, and

knowledge of how to conduct research are all skills that I greatly admire, and he really

helped me get a grip on the research process. I am also very grateful to a number of

other students and researchers with whom I’ve had the opportunity to work closely.

I’d like to thank Na Young Kim for her interest and excitement for the silicon project,

as well as for her organizational skills. I very much enjoyed working with her, and

hope she will keep the project going. I’d like to thank Susan Clark and Kaoru Sanaka,

who I worked closely with on the zinc selenide project. I had a great time working

many hours in the lab with Susan and am thankful for her contagious enthusiasm

for the project. I enjoyed working with Kaoru as well, whose determination for the

vi

project I admire.

I’d like to thank Kristiaan de Greve and Peter McMahon for their useful discus-

sions and enjoyable distractions in lab, as well as Katsuya Nozawa and Tomoyuki

Horikiri for their help and guidance in getting the silicon project started. I would

also like to thank the other members of the Yamamoto group, present and past,

all of whom have contributed to my time here, either through direct research, or

through fruitful and enjoyable interactions: Leo Yu, Zhe Wang, Cody Jones, Wolf-

gang Nitsche, Shruti Puri, Kai Wen, David Press, Georgios Roumpos, Kai-Mei Fu,

Qiang Zhang, Shinichi Koseki, Chandra Natarajan, Shelan Tawfeeq, Jung-Jung Su,

and many others.

I am very thankful for the chance to work with Alex Pawlis, through our zinc

selenide collaboration. The samples he grew gave us the best opportunity for devel-

oping the system, and his understanding of the system and continued improvement

of the growth process were crucial. I’d like to thank Michael Thewalt for his deep

insight and understanding into the phosphorus donor system in silicon, and to Sven

Rogge for his devices and for collaborating with us on the preliminary silicon device

experiments.

Many thanks to the numerous people who helped keep the Yamamoto group,

Ginzton, and the physics department running smoothly. I am especially very grateful

to Yurika Peterman and Rieko Sasaki for all their help in running the Yamamoto

group, I always enjoyed the opportunity to visit their office and talk with them.

I am thankful to Maria Frank for running the physics office and handling all the

paperwork and requirements, and to the Ginzton office for approving all the liquid

helium orders and other requests necessary to keep the experiments running. I’m also

very grateful to Mike Schlimmer for keeping everything in the building running well

and for helping solve building issues, and to Larry Randall for all his help solving

computer and electronics issues.

I am so very thankful to all my friends, at Stanford and elsewhere, for keeping me

going through their continued support, distractions, and great times. I have been very

fortunate to meet such a fantastic group of friendly people at Stanford, particularly

the guys in the physics program, who have made my time here so enjoyable. While the

vii

number of people who have impacted me are more numerous than can reasonably be

listed here, I’d like to thank three guys in particular, Dan Walker, Phil Van Stockum,

and John Ulmen, who I had the opportunity to live with and who have been a part

of the vast majority of my best memories at Stanford.

Finally, I am extremely thankful to my family for all their support. To my parents,

Cathy and Jay, for their support and confidence in me, and for always teaching me

while growing up that I need to balance working hard with playing hard. My father’s

insistence that I never ‘baby the equipment’ has given me the confidence to push the

limits, and my mother’s reason and kindness has kept me (moderately) levelheaded.

To my siblings, Bryan and Kristi, and Lauren and Kevin, thank you for keeping me

down to earth and for all the fun times we had together when we’re able to get away

from work. And last, but not least, to my best friend and girlfriend, Jackie, thank

you for giving me a reason and a purpose to finish grad school and I can’t wait to

start on our next adventure together.

viii

Contents

Abstract v

Acknowledgements vi

1 Introduction 1

1.1 Quantum bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Qubit requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Quantum key distribution . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Quantum repeaters . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Qubit candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Neutral donor system 13

2.1 Effective mass theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Donor-bound exciton state . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Optical transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Nuclear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Spin qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Fluorine donors in ZnSe 27

3.1 Single donor isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Optical spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Magnetophotoluminescence . . . . . . . . . . . . . . . . . . . 34

3.2.2 Single photon source . . . . . . . . . . . . . . . . . . . . . . . 35

ix

3.3 Single donor confirmation . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.1 Time averaged optical pumping . . . . . . . . . . . . . . . . . 44

3.4.2 Time resolved optical pumping . . . . . . . . . . . . . . . . . 48

3.5 F:ZnSe outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Phosphorus donors in Si 55

4.1 P:Si optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Photoluminescence excitation spectroscopy . . . . . . . . . . . . . . . 59

4.3 Zero-field splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.1 Strain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.2 Fitting Results . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Electrical detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.1 Quantum Hall charge sensor . . . . . . . . . . . . . . . . . . . 67

4.4.2 Description of device and measurement scheme . . . . . . . . 68

4.5 Device physics and simulation . . . . . . . . . . . . . . . . . . . . . . 71

4.5.1 Donor electron ground state . . . . . . . . . . . . . . . . . . . 72

4.5.2 Edge Channel Scattering . . . . . . . . . . . . . . . . . . . . . 78

4.5.3 Ionization and recapture . . . . . . . . . . . . . . . . . . . . . 82

4.5.4 Optical transition . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5.5 Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . . 85

4.5.6 Device prospects . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.6 Preliminary experiments with FinFET device . . . . . . . . . . . . . 86

4.6.1 Device structure and behavior . . . . . . . . . . . . . . . . . . 86

4.7 P:Si outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Conclusion and Outlook 91

A Selection rules 94

A.1 F:ZnSe selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

A.2 P:Si selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

x

B Experimental setup 106

B.1 F:ZnSe experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

B.2 P:Si experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography 111

xi

List of Tables

1.1 Comparison between three qubit candidates . . . . . . . . . . . . . . 11

4.1 Lifetimes of decay mechanisms in P:Si . . . . . . . . . . . . . . . . . 57

4.2 Fitting parameters for the strain model . . . . . . . . . . . . . . . . . 63

4.3 Energy fitting parameters . . . . . . . . . . . . . . . . . . . . . . . . 66

xii

List of Figures

1.1 Representation of possible cbit and qubit states . . . . . . . . . . . . 3

1.2 Diagram of the BB84 QKD protocol . . . . . . . . . . . . . . . . . . 5

1.3 Function of a quantum repeater . . . . . . . . . . . . . . . . . . . . . 7

2.1 Substitutional donor . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Neutral donor system . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Lambda system used for optical control . . . . . . . . . . . . . . . . . 26

3.1 F:ZnSe sample structure . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Example bulk F:ZnSe spectra . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Example 4 nm quantum well F:ZnSe spectra . . . . . . . . . . . . . . 32

3.4 Example 2 nm quantum well F:ZnSe spectra . . . . . . . . . . . . . . 33

3.5 D0X transition at 7 T . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Hanbury-Brown-Twiss experiment . . . . . . . . . . . . . . . . . . . . 36

3.7 Example g2(τ) data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Hong-Ou-Mandel experiment . . . . . . . . . . . . . . . . . . . . . . . 38

3.9 Spectra of a confirmed single donor . . . . . . . . . . . . . . . . . . . 39

3.10 Power saturation of a confirmed single donor . . . . . . . . . . . . . . 40

3.11 Zeeman splitting of a confirmed single donor . . . . . . . . . . . . . . 41

3.12 g2(τ) of a confirmed single donor . . . . . . . . . . . . . . . . . . . . 42

3.13 Diagram of the optical pumping scheme . . . . . . . . . . . . . . . . 44

3.14 Resonant and power dependent optical pumping behavior . . . . . . . 47

3.15 Time-dependent optical pumping behavior . . . . . . . . . . . . . . . 48

3.16 Representation of the Monte-Carlo simulation . . . . . . . . . . . . . 49

xiii

4.1 P:Si optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Zero-field splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Magnetic field dependent transition energies . . . . . . . . . . . . . . 65

4.4 Measurement device schematic . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Quantum Hall effect within the measurement device . . . . . . . . . . 69

4.6 Eigenstate energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.7 Potential energy countour plot . . . . . . . . . . . . . . . . . . . . . . 76

4.8 Edge state scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.9 FinFET device schematic . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.10 Temperature dependent I-V curves . . . . . . . . . . . . . . . . . . . 88

4.11 FinFET switching behavior . . . . . . . . . . . . . . . . . . . . . . . 89

A.1 Hole wavefunction in a quantum well . . . . . . . . . . . . . . . . . . 95

A.2 F:ZnSe Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.3 P:Si Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.4 High-field P:Si spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.1 F:ZnSe experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 107

B.2 P:Si experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 109

xiv

Chapter 1

Introduction

Quantum information is the relatively new field which uses the mathematics of quan-

tum mechanics to perform mathematical operations that are not available when using

the mathematics of classical mechanics. As the computers used today (classical com-

puters) use physical classical states to represent mathematical numbers and classical

interactions to perform mathematical operations, quantum computers use physical

quantum states to represent mathematical numbers and quantum interactions to per-

form mathematical operations. While large-scale quantum computers do not yet exist,

researchers have discovered algorithms for quantum computers that can solve certain

problems much faster than is possible on a classical computer [1, 2, 3, 4].

In addition to enabling quantum computers, quantum mechanics can be used to

distribute cryptographic keys securely (quantum key distribution) [5], and to simu-

late the behavior of complex quantum interactions in one system using simpler and

well-understood quantum interactions in another system (quantum simulator) [1].

The potential benefits of quantum simulators, quantum key distribution, and quan-

tum computers currently drive a large amount of international research into various

types of quantum systems to see which system can most easily be used as the basic

representation of quantum information.

1

2 CHAPTER 1. INTRODUCTION

1.1 Quantum bits

The simplest representation of quantum information is a two-state quantum system,

where one state represents the value 0 and the other state represents the value 1. In

analogy with the classical bit (cbit), such systems are called quantum bits, or qubits.

The main difference between classical bits and quantum bits is that while a classical

bit can only be in state 0 or state 1, the quantum bit can be in state |0〉, state |1〉, ora superposition of both states α |0〉+ β |1〉.

This superposition state can be thought of as what makes quantum information so

powerful. In particular, it enables “quantum parallelism”, where a single computation

can be performed on a superposition of all possible discrete inputs at the same time.

In classical computation, only one input can be sent through the processor at a time.

However, obtaining the result of a particular computation is a bit more complex in

a quantum computer. A measurement of the final state of the qubits generally gives

the result of the computation of one input value rather than the result of all the

inputs. Fortunately, due to the wave nature of quantum mechanics, the inputs can

be superimposed in a way that the final states constructively interfere to produce

the desired answer. This is how some of the known quantum algorithms are able to

compute the answer.

The cbit is often described by a two-state switch, either off or on (Fig. 1.1(a)).

The qubit, on the other hand, has an infinite number of possible superposition states.

Its SU(2) symmetry implies that any qubit state can be described as

|ψ〉 = cos (θ/2) + eiφ sin (θ/2) , (1.1)

where θ and φ are the two degrees of freedom available to the state. This equation

describes the surface of a sphere, which leads to the description of a qubit state by

a position on the so-called Bloch Sphere. As shown in Fig. 1.1(b), the north pole

represents the state |1〉, the south pole represents the state |0〉, and any other locationon the surface of the sphere represents a superposition state.

The qubit is not the only possible basic building block of a quantum computer.

Other systems containing three or more states have been proposed, such as the qutrit

1.2. QUBIT REQUIREMENTS 3

Figure 1.1: Representation of the possible states for (a) cbits, and (b) qubits.

(three-level system) or qudit (n-level system). These other quantum information

building blocks are essentially just as useful as qubits, but are more complex to work

with. This thesis will focus only on qubit implementations.

1.2 Qubit requirements

The first description of the necessary requirements in order for a qubit to be a prac-

tical basis for a quantum computer was suggested by David DiVincenzo [6]. These

requirements are:

1) a scalable physical system with well characterized qubits,

2) the ability to initialize the state of the qubits,

3) decoherence times much longer than the gate operation time,

4) a universal set of quantum gates, and

5) a qubit-specific measurement capability.

These are often supplemented by a variety of more specific criterion for particular

qubit implementations. However, these criteria are a good starting place, and have

yet to be fully met by any qubit candidate system. There are many qubit candidate

systems currently being studied, from single photon states [7] to individual ions [8] to

superconducting currents [9]. Each candidate fits differently with this set of criteria,

and there is no clear best candidate at the moment. For instance, the first successful


quantum algorithm was performed using nuclear magnetic resonance (NMR) on an

ensemble of molecules where the nuclear spins within each molecule served as the

quantum bits [10]. In this system, criteria (2)-(5) were met, but the lack of scalability

(more bits would require increasingly large molecules, each with different resonances)

makes it an unreasonable candidate system. Other systems, such as semiconductor

systems, have found scalability to be easy, but two-qubit gates and decoherence time

have been difficult to achieve. However, each of these criteria are required in order to

have a large-scale quantum computer.

1.3 Quantum key distribution

In addition to the benefits of quantum computation, quantum mechanics also pro-

vides a method to create shared information between two separated locations through

a quantum channel while making it impossible (according to the laws of quantum

mechanics) for a third party to intercept this information undetected. This is called

quantum key distribution (QKD), and generally places less stringent requirements on

qubit systems than required for quantum computation.

In its simplest implementation, QKD requires a well defined qubit that can be

initialized and measured and can be easily transported between two locations, called

a flying qubit. QKD systems already exist and are even sold commercially. These

systems use photon states for the qubit basis, and transport the qubits through optical

fibers between two locations. One particular implementation I will describe here uses

the polarization of single photons as the qubit states. This protocol is called BB84 [5].

The BB84 system consists of an optical fiber between two parties, who we can call

Alice and Bob. Alice encodes a random string of classical bits into the polarization

of single photons that she creates and sends to Bob through the fiber. For each

photon qubit, she randomly chooses the basis she uses to encode the classical bit,

selecting either rectilinear (horizontal |H〉 & vertical |V 〉), or diagonal (diagonal |D〉& antidiagonal |A〉). If she chooses rectilinear, then she encodes the classical bitaccording to |H〉 = 0, |V 〉 = 1. If she chooses diagonal, she encodes the bit accordingto |D〉 = 0, |A〉 = 1 (see Fig. 1.2).

1.3. QUANTUM KEY DISTRIBUTION 5

Figure 1.2: Diagram representing the BB84 QKD protocol. Single photons are sentdown an optical fiber from Alice to Bob. Alice encodes each bit in one of the twopolarization bases, randomly selected. Bob detects the photons he receives in one ofthe polarization bases, again randomly selected.

Next, Bob detects each photon he receives using a random basis. Following this,

Alice and Bob compare the creation and measurement basis for each photon over an

open classical channel. If Bob detects a photon in the same basis in which it was

created (50% of the time on average), he measures the same bit as was encoded by

Alice. However, if he chooses a different basis from which it was created, he measures

a random bit value compared to what was encoded. Alice and Bob keep the bit values

of the photons which were created and measured in the same basis as their new shared

cryptographic key, and discard the other bits.

Since Alice and Bob only share the creation and measurement basis over the

open classical channel, they transmit no information about the actual shared key bit

values that an eavesdropper, Eve, could use to learn anything about the key. On the

other hand, if Eve listens in on the quantum channel (the optical fiber), she will be

unable to gain any information without modifying the quantum states of the photon

qubits. Due to the quantum no-cloning theorem, she is unable to make copies of

the single photon states, and her only options are to attempt to detect the photon

polarization without destroying the photon, or detect the photon and then send a

new photon with the same state as she detected. Eve does not know the basis in

which the photons were created, and so 50% of the time she measures in the wrong


basis, gaining no information. Furthermore, each time she measures in the wrong

bases, she unavoidably modifies the quantum state sent to Bob.

Any modification of the quantum states between Alice and Bob can easily be

detected by comparing some of the shared bits over the open classical channel. On

average, 25% of the bits Eve attempts to detect will introduce an error in Bob’s bits.

Thus, if Alice and Bob determine they have an unusually large error rate, they know

someone must be attempting to listen. However, if the error rate is normal, they know

that they have a completely secure shared bit string which they can use to encrypt

information over a classical channel.

The quantum key distribution networks that exist today all rely on the transmis-

sion of photons in optical fibers. While the systems work, they all share one limitation:

lossy fibers. Classical optical networks get around this using repeaters, where an op-

tical signal is periodically amplified in order to overcome any loss. Unfortunately,

since quantum mechanics forbids the copying of quantum states, we cannot amplify

a quantum signal. For this reason, current QKD systems are limited to 100-200 km

transmission distances for reasonable bitrates due to the loss of the single-photon

flying qubits. Longer transmission distances will require the development of a new

type of repeater.

1.3.1 Quantum repeaters

Quantum repeaters are different from classical repeaters in that they do not amplify

a signal. Instead, they serve to build up entangled resources over long distances,

which can then be used to transfer qubit states from one location to another through

quantum teleportation. The qubit requirements for quantum repeaters are greater

than that required for general QKD, and in fact requires all of the elements listed in

Sec. 1.2 for general quantum computers. However, the scale of the system required for

quantum repeaters is usually considered to be much less than that of a full quantum

computer, and so is a good intermediate technological goal, somewhere in between

the simple QKD protocols and a full quantum computer system.


Figure 1.3: Schematic of how a quantum repeater would work. (a) Alice, Bob, andthe Repeater all send single photons entangled with their stationary qubits throughoptical fibers. (b) These states can be re-written as a sum of Bell states betweenthe stationary qubits and between the flying qubits. After photon detection, thestationary qubits are projected into one of the Bell states. (c) The quantum repeaterperforms entanglement swapping by performing a Bell-state measurement of its twostationary qubits, projecting Alice and Bob’s qubits into an entangled Bell state.

Quantum repeaters require two types of qubits: flying qubits (such as single pho-

tons, as used in QKD), and stationary qubits. The flying qubits are used to transfer

quantum information between repeaters, while the stationary qubits are used to store

the quantum information. The protocol works as follows (shown in Fig. 1.3).

Alice and Bob both have a stationary qubit which they entangle to a flying qubit,

which we will assume is a single photon. Thus, Alice and Bob start with qubit states

|ΨA〉 =1√2

(|0〉1 |H〉1 − |1〉1 |V 〉1) , (1.2)

|ΨB〉 =1√2

(|0〉4 |H〉4 − |1〉4 |V 〉4) , (1.3)


where |0〉 and |1〉 refer to the stationary qubit, and |H〉 and |V 〉 refer to the flyingqubit. A quantum repeater between Alice and Bob has two pairs of stationary and

flying qubits

|ΨR1〉 =1√2

(|0〉2 |H〉2 − |1〉2 |V 〉2) , (1.4)

|ΨR2〉 =1√2

(|0〉3 |H〉3 − |1〉3 |V 〉3) . (1.5)

Alice and Bob’s flying qubits fly towards the repeater, while the repeater’s flying

qubits fly towards Alice and Bob (Fig. 1.3(a)). At this point, the states can be

rewritten as

|ΨA〉 |ΨR1〉 =1

2(|0〉1 |H〉1 − |1〉1 |V 〉1) (|0〉2 |H〉2 − |1〉2 |V 〉2) , (1.6)

=1

4[(|0〉1 |0〉2 + |1〉1 |1〉2) (|H〉1 |H〉2 + |V 〉1 |V 〉2)

+ (|0〉1 |0〉2 − |1〉1 |1〉2) (|H〉1 |H〉2 − |V 〉1 |V 〉2)

− (|0〉1 |1〉2 + |1〉1 |0〉2) (|H〉1 |V 〉2 + |V 〉1 |H〉2)

− (|0〉1 |1〉2 − |1〉1 |0〉2) (|H〉1 |V 〉2 − |V 〉1 |H〉2)] , (1.7)

=1

2

(∣∣Φ+〉12

∣∣φ+〉12

+∣∣Φ−〉

12

∣∣φ−〉12

−∣∣Ψ+〉

12

∣∣ψ+〉12−∣∣Ψ−〉

12

∣∣ψ−〉12

), (1.8)

where the capital Ψ and Φ refer to the Bell states for the stationary qubits, and the

lower-case ψ and φ refer to the Bell states for the flying qubits. The qubits states

between the repeater and Bob can be described in the same way,

|ΨB〉 |ΨR2〉 =1

2

(∣∣Φ+〉34

∣∣φ+〉34

+∣∣Φ−〉

34

∣∣φ−〉34

−∣∣Ψ+〉

34

∣∣ψ+〉34−∣∣Ψ−〉

34

∣∣ψ−〉34

). (1.9)

The photons from Alice and the repeater meet in the middle of the optical fiber on

a beamsplitter, and will travel along paths 1’ and 2’ towards single-photon counters


(Fig. 1.3(b)). After the beamsplitter, the four flying-qubit Bell states transform into

∣∣φ+〉12→ 1

2([|HH〉1′ + |V V 〉1′ ] |0〉2′ − |0〉1′ [|HH〉2′ + |V V 〉2′ ]) (1.10)∣∣φ−〉

12→ 1

2([|HH〉1′ − |V V 〉1′ ] |0〉2′ − |0〉1′ [|HH〉2′ − |V V 〉2′ ]) (1.11)∣∣ψ+〉

12→ 1√

2(|HV 〉1′ |0〉2′ − |0〉1′ |HV 〉2′) (1.12)∣∣ψ−〉

12→ 1√

2(|V 〉1′ |H〉2′ − |H〉1′ |V 〉2′) , (1.13)

where |P1P2〉 refers to two photons in a particular mode with the specified polariza-tions P1 and P2 (H or V ) and |0〉 refers to zero photons in a particular mode. Inprinciple, all four of these states are distinguishable. In practice, using only two po-

larization and number insensitive detectors, as shown in Fig. 1.3, only the |ψ−〉 statecan be distinguished (|ψ+〉 can also be distinguished if four detectors are used). |ψ−〉is the only state that has a photon on each path, so a coincidence of detection events

in both detectors is a measurement of the photon state |ψ−〉. This then projects thestationary qubits into the entangled state |Ψ−〉.

A beam splitter and pair of detectors between Bob and the repeater perform the

same entanglement procedure on their side. This procedure can be repeated on each

side until a coincidence detection event indicates a successful entanglement. Once

both Alice and Bob have their stationary qubit entangled with a repeater qubit, the

entanglement swapping phase begins. At the start of this phase, the four stationary


qubit states can be written as

∣∣Ψ−〉12

∣∣Ψ−〉34

=1

2(|0〉1 |1〉2 − |1〉1 |0〉2) (|0〉3 |1〉4 − |1〉3 |0〉4) , (1.14)

=1

4[− (|0〉1 |0〉4 + |1〉1 |1〉4) (|H〉2 |H〉3 + |V 〉2 |V 〉3)

+ (|0〉1 |0〉4 − |1〉1 |1〉4) (|H〉2 |H〉3 − |V 〉2 |V 〉3)

− (|0〉1 |1〉4 + |1〉1 |0〉4) (|H〉2 |V 〉3 + |V 〉2 |H〉3)

− (|0〉1 |1〉4 − |1〉1 |0〉4) (|H〉2 |V 〉3 − |V 〉2 |H〉3)] , (1.15)

=1

2

(−∣∣Φ+〉

14

∣∣Φ+〉23

+∣∣Φ−〉

14

∣∣Φ−〉23

+∣∣Ψ+〉

14

∣∣Ψ+〉23

+∣∣Ψ−〉

14

∣∣Ψ−〉23

). (1.16)

The quantum repeater next measures its two stationary qubits (2 & 3) in the Bell

state basis. This can be accomplished, for instance, by applying a CNOT 2-qubit

gate followed by a Hadamard 1-qubit gate and a measurement in the {|0〉 , |1〉} basis.When this occurs, Alice and Bob’s qubits (1 & 4) are projected into an entangled

Bell state (Fig. 1.3(c)). This procedure is called entanglement swapping, where the

entanglement between qubits 1 & 2 is swapped for entanglement between qubits 1 &

4 by consuming the entanglement between qubits 3 & 4.

Now that Alice and Bob have an entangled qubit pair, they can simply measure

their own qubit in the {|0〉 , |1〉} basis. Based upon the information of the measuredstate of the repeater qubits (which the repeater sends to Alice and Bob over an open

classical channel), Alice and Bob know the measured value of the other person’s

qubit without communicating any information. After running this entire procedure

numerous times, they will have built up a sequence of shared bits that only they

know. As with the simple QKD scheme from Sec. 1.3, Alice and Bob can openly

share a few of these bits to check their error rate and determine if a third party was

attempting to get in the middle of their entanglement procedure.

Once a quantum repeater can be built, multiple repeaters can be chained to-

gether in series to extend the range of quantum key distribution essentially without

limit. Since the entanglement creation along individual optical paths can be done

asynchronously and in parallel, the time to create entangled pairs between Alice and

1.4. QUBIT CANDIDATES 11

Bob will depend upon the length of the individual segments instead of on the total

distance between them (assuming the entanglement swapping procedure takes negli-

gible time). Quantum repeater technology will be critical in enabling long-distance

quantum key distribtution.

1.4 Qubit candidates

There are many candidate systems for qubit implementations currently under inves-

tigation. The best candidates for quantum repeater technology are those that easily

interact with optical photons. Thus, for this application, the leading three candidate

systems at the moment are trapped ions in a vacuum [8, 11, 12], nitrogen vacancies

(NV) centers in diamond [13, 11, 14, 15, 16], and self-assembed InGaAs quantum

dots [17, 18, 19, 11, 20, 21, 22, 23, 24, 25]. Great progress has been made with each

of these systems, however, none of them have met all of the requirements necessary

for a quantum repeater. A summary of eight important requirements and how each

system performs for each is shown in Table 1.1 below.

Trapped ion NV center InGaAs QD

Optical quantumefficency

100% 3% [13] 97% [18]

Homogeneity 1:1 1,000:1 [13] 10,000:1 [19]Decoherencetime

15 s [11] 2 ms [11] 3 µs [11]

Initialization 99% in 10 ms [12] 5 µs [14] 92% in 13 ns [19]1-qubit control 99% in 1 µs [12] 48 ns [14] 94% in 4 ps [19]2-qubitinteraction

Phonon coupling Nuclear spincoupling

Theory

Projectivemeasurement

Cycling, slow Destructive, lowsuccess rate

Destructive, lowsuccess rate

Fabrication &integration

Difficult Difficult, cavityintegrationpossible

Easier, cavityintegrationpossible

Table 1.1: Comparison between the three qubit candidates. Homogeneity is indicatedby the ratio between the ensemble linewidth and the homogeneous linewidth.


Trapped ion systems have perfect optical quantum efficiency in addition to un-

equalled homogeneity and long decoherence times, but qubit control is relatively slow,

and it is very difficult to fabricate and integrate trapped ion systems into deployable

technology due to the addition requirements of cooling and trapping the ions. Ni-

trogen vacancies are reasonably homogeneous, have long decoherence times coupled

with fast gate operations, but have a low quantum efficiency and it is very difficult

to fabricate devices made out of diamond. InGaAs QDs, on the other hand, have

high quantum efficiency, very fast control times, and devices are easy to fabricate,

but they suffer from inhomogeneity and decoherence, and have not yet achieved any

sort of two-qubit gate experimentally.

In order for a qubit system to enable quantum repeater technology, it must score

well in each of these categories. While a great deal of progress has been made in

reaching this goal since the idea of quantum key distribution was first presented, each

system still has a lot more that needs to be accomplished.

1.5 Thesis outline

This thesis will focus on a fourth qubit candidate system, which has the potential

to avoid some of the biggest obstacles impeding progress in the other systems. This

system is the neutral donor system. Chapter 2 will introduce the neutral donor system

and discuss the important characteristics that make it a good qubit system. Chapter

3 will present fluorine donors in zinc selenide as a particularly good candidate for

quantum repeater technology and present our experimental results on the optical

pumping of a single fluorine donor for qubit initialization. Chapter 4 will present

phosphorus donors in silicon as a good candidate for quantum computer technology

and present theoretical results on a technique for measuring a single phosphorus

donor, as well as discuss some related preliminary experimental results. Chapter 5

will then conclude and present an outlook for neutral donor systems.

Chapter 2

Neutral donor system

Isolated semiconductor spins are natural qubits due to their well defined quantum

states. These individual spins can be isolated in a number of ways, but most of the

methods boil down to creating a local potential energy minimum which can localize

the spin, and making the energy levels of the spin system distinguishable from the

energy levels of neighboring spins. The most common isolation systems are quantum

dots, electric gates, and semiconductor impurities. The focus of the remainder of this

thesis will be on one particular type of semiconductor impurity, a single donor.

A substitutional donor in a semiconductor crystal provides a single electron which

is located in the conduction band at room temperature. However, the donor creates

a shallow attractive potential which at sufficiently low temperatures can trap that

electron in a localized state just below the conduction band edge. This is referred

to as a donor-bound electron, or a neutral donor state D0 (Fig. 2.1). Effective

mass theory allows us to compute the energy of these states and approximate the

wavefunction of the bound electron, as will be shown in Sec. 2.1.

In addition to an electron, the donor can also trap an exciton, a quasiparticle

composed of an electron and a hole bound together. This excited state is called the

donor-bound exciton state D0X (Sec. 2.2). Optical transitions between D0 and D0X

states can be used to interact with and manipulate the donor-bound electron spin.

13

14 CHAPTER 2. NEUTRAL DONOR SYSTEM

Figure 2.1: A substitutional donor within the semiconductor crystal with a singlebound electron.

2.1 Effective mass theory

A semiconductor crystal is a complicated system, involving many interacting elec-

trons and nuclei. Fortunately, effective mass theory [26, 27, 28] allows us to greatly

simplify the Hamiltonian of the system in order to predict the behavior of electrons in

particular states. The full Hamiltonian for a defect-free crystal and a single electron

can be described by the equation

H0 = −h̄2

2m0∇2 + Vp, (2.1)

where m0 is the free electron mass and Vp is the periodic potential of the semicon-

ductor crystal lattice.

To apply effective mass theory, we first use Bloch’s theory to rewrite the wave-

function in the form of a sum of Bloch waves times an envelope function,

Ψ =∑k,n

ak,neik·rψk,n(r), (2.2)

where ψk,n(r) is a periodic Bloch function that has a periodicity equal to that of Vp.

2.1. EFFECTIVE MASS THEORY 15

Due to the linear nature of the Hamiltonian, we can go through the following deriva-

tion using just one particular Bloch wavefunction, and the results will be accurate for

a sum of Bloch wavefunctions. The Bloch wavefunction form allows us to separate

the component of the Hamiltonian describing the periodic potential from the other

components that are of interest. Then,

H0Ψ =

(− h̄

2

2m0∇2 + Vp

)eik·rψk,n(r) (2.3)

=

(− h̄

2

2m0∇2eik·r

)ψk,n(r) + Ep,i(k)Ψ, (2.4)

where

Ep,i(k)Ψ =

(− h̄

2

2m0

(−2ik · ∇ψk,n(r) +∇2ψk,n(r)

)+ Vpψk,n(r)

)eik·r. (2.5)

Now, all of the complication of the periodic potential is contained within Ep,i(k),

which depends upon both the momentum k and the band index n. For many semi-

conductors, this energy has a minimum around k = 0, allowing us to approximate it

as

Ep,n(k)Ψ ≈(En + αnk

2)

Ψ. (2.6)

We can then rewrite the full Hamiltonian as

H0Ψ = EnΨ +

(− h̄

2

2m0∇2eik·r + αik2eik·r

)ψk,n(r) (2.7)

= EnΨ +

(− h̄

2

2m∗∇2eik·r

)ψk,n(r), (2.8)

where we combine the term describing the momentum of the free-particle wave with

the term describing the 2nd order approximation of Ep,n(k), giving the system an

effective mass m∗. Now all of the effect of the periodic crystal potential is contained

within En and m∗, and the Bloch wave can be ignored, resulting in an effective

Hamiltonian Heff and and an envelope wavefunction Ψeff for a particular band n


(assuming k is near the band minimum),

HeffΨeff =

(En −

h̄2

2m∗∇2)

Ψeff, (2.9)

Ψeff =∑k

akeik·r. (2.10)

The full Hamiltonian for an electron in a semiconductor crystal has now been

reduced to the Hamiltonian for a free particle of mass m∗ and a base energy En.

Eeffective mass theory has allowed us to greatly simplify the interactions of an elec-

tron. It is important to recognize that m∗ is not the real mass of the electron, but

determines the relationship between the crystal momentum of the electron k and its

energy when k is near the band minimum. Furthermore, we have assumed that the

electrons are non-interacting. This is most valid at the top of the valence band and

in the conduction band, and those are the only electron states we will be considering.

Note that we performed this derivation under the assumption that the band minimum

was at k = 0, but in fact a similar derivation can be performed when the minimum

is not at k = 0.

Next, we can add in the potential VD created by replacing a crystal nucleus with

the donor nucleus. We will only be considering the lowest-energy conduction band

and the highest-energy valence band, and so in this case, the screening of the electrons

in the inner-most valence bands or shells causes the potential of the donor nucleus to

resemble that of a single positive charge,

VD = −e2

4π�r, (2.11)

where � is the static dielectric constant of the semiconductor.

The effective Hamiltonian for an electron in the conduction band in the presence

of a donor is now

Heff = Ec −h̄2

2m∗∇2 − e

2

4π�r, (2.12)

which resembles the Hamiltonian for a hydrogen atom. In fact, the solutions are

identical after the substitution of m∗ for m0, � for �0, and adding the energy of the

2.2. DONOR-BOUND EXCITON STATE 17

conduction band minimum Ec. This tells us that a donor provides localized electron

states with envelope functions resembling hydrogen wavefunctions:

Ψeff = ψnlm(r, θ, φ), (2.13)

E = Ec +1

n2e4m∗

2h̄2(4π�)2. (2.14)

The electron also has an effective Bohr radius,

a∗ =h̄24π�

m∗e2. (2.15)

The lowest energy state, where n = 1, l = 0, and m = 0, is generally referred to as

the neutral donor state D0.

Effective mass theory in fact turns out to give a reasonable approximation for the

binding energy of an electron in many systems (the difference in energy between the

conduction band edge and the donor-bound electron state). However, the approxi-

mation does not accurately describe the interactions very near to the donor where

the inner-shell electrons exist, and so effective mass theory is not exact. A correction

term is needed to account for the difference in energy between the experimentally

measured energy and the energy predicted by effective mass theory. This correction

is called the central cell correction [27], and is different for each donor system. The

central cell correction is large for so called deep donors, which have a large binding

energy and therefore a significant portion of the electron wavefunction overlaps with

the central cell region in the immediate proximity of the donor. Shallow donors, on

the other hand, have a small central cell correction and are therefore well described

by effective mass theory. Both fluorine in zinc selenide and phosphorus in silicon are

considered shallow donors, and are well approximated by effective mass theory.

2.2 Donor-bound exciton state

The donor-bound exciton state D0X is composed of an exciton bound to a neutral

donor. An exciton is a quasiparticle formed when an electron in the conduction band


and a hole in the valence band bind to one another, as shown in Fig. 2.2. The hole

itself is a quasiparticle, resulting from the cooperative behavior of the electrons in a

valence band missing an electron, but can be described as a particle with a charge

of +e, an effective mass of m∗h, and a spin opposite that of the unoccupied valence

band electron state. The exciton is a bound state between an electron and a hole,

and can also be approximated by effective mass theory [29]. Since the exciton is a

bound state, the energy to create it is slightly less than the energy required to take an

electron from the valence band and put it into the conduction band. The difference

between these energies is the exciton binding energy EX. The exciton also has an

effective radius, which for semiconductor systems is generally spread over many unit

cells and as a Wannier exciton [30].

Figure 2.2: Diagram of the D0 ground state with a single bound electron, and theD0X state with an additional electron-hole pair. An optical transition connects thetwo states.

Most excitons are free excitons, and are able to move around within the semi-

conductor crystal. A neutral donor, however, provides a potential which can bind

a single exciton. In this state, two electrons and a hole are both localized around a

single donor. It might seem unclear why a neutral particle to another neutral particle,

2.2. DONOR-BOUND EXCITON STATE 19

but it could be described as analogous to how two atoms with unpaired electrons will

form a covalent bond by sharing electrons.

Computation of the binding energy of an exciton to a neutral donor is a very

difficult task due to the 4-body nature of the state. However, an experimental relation

called Hayne’s rule [31] has been very successful in predicting D0X binding energies.

Hayne’s rule says that the binding energy of the exciton to a neutral donor should be

proportional to the binding energy of the electron to the donor in the D0 state,

ED0X ≈ aED0 , (2.16)

where a is the proporionality constant and is generally different for each semiconduc-

tor.

The two electrons in the D0X state form a spin singlet in order to exist in the

same spatial wavefunction state, and so the hole determines the spin of the D0X

state. Since the electrons are in the bottom of the conduction band, their Bloch

wavefunctions share the symmetry of the s-orbitals for atomic electrons, with zero

units of orbital angular momentum. The holes, on the other hand, are at the top

of the valence band, and share the symmetry of p-orbitals, with one unit of orbital

angular momentum. Therefore, spin-orbit coupling determines the effective spin of

the hole states by combining the 1/2 unit of spin angular momentum with the one

unit of orbital angular momentum.

The total spin 3/2 manifold is higher in the valence band than the total spin

1/2 due to the energy of the spin orbit coupling, and therefore a total spin 3/2 hole

requires less energy to create. This means that the total spin 1/2 D0X states are

noticeably higher in energy than the the total spin 3/2 D0X states, and can often

be unbound states that play no role in the system. The total spin 3/2 manifold is

broken up into spin projection ±3/2 and ±1/2 states. The ±3/2 states have a largereffective mass than the ±1/2 states, and therefore the former are called heavy-hole(HH) states while the latter are called light-hole (LH) states.

The D0X state can be created by bringing a valence electron into the conduction

band. The electron can then bind with the hole it left behind, creating an exciton,


and then the exciton can further bind to the neutral donor. Thus the energy of the

transition between the D0 state and the D0X state is given by

E(D0 → D0X) = EG − EX − ED0X, (2.17)

where EG is the band gap between the minimum in the conduction band and the

maximum in the valence band. In general, the D0 → D0X transition energy is justbelow that of the band gap, which is in the optical energy range. For many semi-

conductors the D0X state can be created resonantly by the absorption of a photon

of the correct energy, assuming spin selection rules are followed and momentum is

conserved.

2.3 Optical transitions

Both the D0 and D0X states are composed of a spatial wavefunction and a spin state:

|Ψ〉 = |ψ(r)〉 |s〉 . (2.18)

Neglecting the nuclear spin for now, there are two D0 states, corresponding to the two

electron spin states |↑〉 and |↓〉. The relative energy of these two states is determinedby the Zeeman splitting due to an applied magnetic field B,

∆E = 2geµBS ·B, (2.19)

where ge is the electron g-factor and µB is the Bohr-magneton.

The D0X states, however, consist of four states, corresponding to the two heavy

hole spin states |±3/2〉 and the two light hole spin states |±1/2〉. Due to spin orbitcoupling, the spatial wavefunction and the spin state are entangled, and so the relative

energies between these states depend on perturbations to the spatial wavefunction in

addition to the Zeeman splitting. Two such perturbations important for this thesis

are strain in the crystal structure, and confinement due to a quantum well.

We can calculate which D0 → D0X transitions are allowed and the relative rates

2.3. OPTICAL TRANSITIONS 21

of these transitions by looking at the electric dipole matrix element between the

states. Fermi’s Golden Rule [32] tells us that the rate of transition for an oscillating

perturbation is proportional to the square of the inner product of the perturbing

Hamiltonian H ′ between the initial and final states:

Ri→f ∝ |〈Ψf |H ′ |Ψi〉|2 . (2.20)

For the case of stimulated optical transitions of electrons, the perturbing Hamiltonian

is the electric dipole operator

H ′ = er · E (2.21)

where E is the complex electric field vector amplitude. In the case of a transition

between D0X and D0, we have

RD0→D0X ∝ |eE · 〈ΨD0X| r |ΨD0〉|2 . (2.22)

All of the D0 states share the same 1s envelope function |φ100〉 and s-like Blochwavefunction |ψ∗n00〉, and can be written as∣∣∣∣ΨD0 ,±12

〉=

{|φ100〉 |ψ∗n00〉 |↑〉|φ100〉 |ψ∗n00〉 |↓〉

, (2.23)

where 100 or n00 indicate the quantum numbers nlm of the hydrogenic wavefunctions

and the ∗ indicates that the Bloch wavefunctions are similar but not identical to the

hydrogenic wavefunctions. In the D0X states, the two electrons form a spin singlet

within the spatial wavefunction given above,

|φ100〉 |ψ∗n00〉1√2

(|↑〉 |↓〉 − |↓〉 |↑〉) , (2.24)

while the hole determines the particular D0X state. The hole envelope function is not

the same as we computed for the electron in D0, especially since it is attracted to the

electrons but repelled from the nucleus. However, like the D0 states, all D0X states

share the same envelope function, which we will notate as |φhh〉. Instead, each hole


state has a unique combination of Bloch wavefunctions:

∣∣∣∣ΨD0X,±32 ± 12〉

=

|φhh〉 |ψ∗n11〉 |↑〉|φhh〉

(√13|ψ∗n11〉 |↓〉+

√23|ψ∗n10〉 |↑〉

)|φhh〉

(√13

∣∣ψ∗n1−1〉 |↑〉+√23 |ψ∗n10〉 |↓〉)|φhh〉

∣∣ψ∗n1−1〉 |↓〉. (2.25)

It is important to note that the inner product in Eq. 2.22 is not an inner product

between the electron in the D0 state and the hole in the D0X state, but is rather

an inner product between the initial and final state of the electron that makes the

jump from the valence band to the vacant donor-bound state. So a transition from∣∣ΨD0 ,+12〉 to ∣∣ΨD0X,+32〉 should be computed by taking the matrix element betweenthe valence band electron state

∣∣ΨD0X,−32〉 (since a negative electron spin correspondsto a positive hole spin) and the donor state

∣∣ΨD0 ,−12〉.In order to compute the inner product between the D0 and D0X states, we need

to know the orientation of the spatial wavefunction with respect to the electric field

of the incident light. In the case of D0, the orientation will be determined by the

characteristics of the semiconductor, such as strain or quantum wells, and is unaffected

by a magnetic field. However, in the case of D0X, where spin-orbit coupling exists, the

orientation can also be effected by an applied magnetic field. Therefore, the allowed

transitions and their respective rates largely will depend upon the characteristics of

the particular semiconductor sample. Computation of the transition rates for F:ZnSe

and P:Si, in particular, are discussed in Secs. A.1 and A.2, respectively.

However, even without detailed calculations, we can gain a bit of insight into which

transitions would be forbidden by noting that the spin components |s〉 commute withr. This means that spin must always be conserved (although total angular momentum

is not necessarily conserved). Conservation of momentum is also required, although

no explicit momentum terms or operators appear in Eq. 2.21.

In a direct bandgap semiconductor, such as ZnSe or GaAs, absorption and emis-

sion of photons is the dominant way to create excitons. In an indirect bandgap semi-

conductor, such as Si, this is generally considered “forbidden” without the emission

2.4. NUCLEAR COUPLING 23

or absorption of an accompanying phonon due to the lack of momentum conserva-

tion. However, a zero-phonon transition is possible in the case of a donor-bound

exciton, even though it is much slower than the non-radiative or phonon-assisted

transitions. The existence of this transition could be thought of as a result of the

momentum-position uncertainty principle. When the exciton is localized to a donor,

it’s momentum becomes more uncertain and the slight overlap between the momen-

tum distribution of the D0X state and the D0 state makes the transition possible.

The mathematics of this momentum overlap would be contained within the Bloch

wavefunctions, which are not explicitly calculated here.

So far we have been considering the lowest energy bound electron states (D0

states). However, bound neutral donor excited states do exist in some circumstances,

which have envelope function that look like 2s or 2p orbitals. Transitions between a

D0X state and an excited neutral donor state D0e have traditionally been called TES

transitions, for two-electron satellite. The name came from the fact that researchers

assumed the transition was associated with two electrons bound to the donor. Tran-

sitions to D0e states have slightly lower energy than transitions to D0 states, and the

difference can be computed using Eq. 2.14. A TES transition can be observed in Fig.

3.2.

2.4 Nuclear coupling

Up until now, we have been ignoring the spin of the donor nucleus. However, in some

semiconductor systems, the nuclear spin plays an important role in determining the

states due to the hyperfine coupling

HHF = AIN · Se. (2.26)


In the absence of a magnetic field, the donor-bound electron spin and the nuclear

spin couple into a set of spin triplets and a spin singlet in the D0 state

|1, 1〉 = |↑e〉 |↑n〉|1, 0〉 = 1√

2(|↑e〉 |↓n〉+ |↓e〉 |↑n〉) , |0, 0〉 = 1√2 (|↑e〉 |↓n〉 − |↓e〉 |↑n〉)

|1,−1〉 = |↓e〉 |↓n〉. (2.27)

However, if the Zeeman splitting (Eq. 2.19) is much stronger than the hyperfine

coupling, then electron and nuclear spin states are decoupled:

|↑e〉 |↑n〉 , |↑e〉 |↓n〉|↓e〉 |↑n〉 , |↓e〉 |↓n〉

. (2.28)

The hyperfine coupling constant A is dominated by the point-contact hyperfine

term for the D0 states since both the s-like Bloch wavefunction and the s-like envelope

wavefunction for the electron overlaps the donor nucleus. However, for the D0X states

with p-like Bloch wavefunctions, the hole wavefunction does not overlap the donor

nucleus and the point-contact term is zero. Since the other terms that contribute to

hyperfine coupling are orders of magnitude smaller than the point-contact term, the

hyperfine coupling between the hole spin in D0X and the nuclear spin is generally

negligible.

Due to the fact that the hyperfine constant A is generally quite small, the hyperfine

split states of D0 are usually not optically resolvable since the width of the optical

transition (determined by the D0X → D0 transition rate) is broader than the thehyperfine splitting. However, as will be discussed in Sec. 4.1, Si is a very interesting

semiconductor since its indirect bandgap produces a very slow transition rate, and the

resulting optical linewidth is far narrower than the hyperfine splitting. This nuclear

distinguishability provides an interface to interact optically with the donor nucleus.

2.5. SPIN QUBIT 25

2.5 Spin qubit

The spin of an electron bound to a donor can serve as a natural qubit by defining

|↑〉 as |0〉 and |↓〉 as |1〉. Semiconductor qubit implementations in general hold manyadvantages over other qubit systems, such as ease of fabrication and scalability, inte-

gration with classical electronics, and a wealth of industry experience working with

those materials. While solid state systems are very complex and provide many chal-

lenges due to many-body interactions, defects, and disorder, a number of experiments

have shown measurement and coherent control of spin qubits in semiconductor de-

vices [33, 34, 35, 25, 36, 37, 38, 39]. In addition to this, donor-bound electrons have

been shown to have high homogeneity and extremely long decoherence times in very

pure systems [40].

The electron spin can be directly manipulated through the use of resonant mi-

crowave radiation. However, manipulation this way is not particularly fast (µs timescale),

and it is difficult to isolate the radiation to one qubit. Therefore, systems which can

be manipulated optically have an immediate advantage in the speed of gate opera-

tions and the scalability of the system due to the strong and localized electric fields

available at optical frequencies. In these interactions, a third, excited state |e〉 isrequired, which is optically connected to the two qubit states. This forms what is

called a lambda-system, as shown in Fig. 2.3.

For electrons trapped in InGaAs quantum dots, the excited state is the trion state,

which is composed of the trapped electron plus an exciton [19]. For donor-bound

electrons, the excited state is the D0X state. Spin rotations have been performed in

both systems utilizing ultrafast two-photon stimulated Raman transitions [41, 19].

These experiments result in ps gate operation times, allowing for orders of magnitude

more gate operations within the decoherence time of the systems.

A great deal of progress has been made on quantum dot qubits (Sec. 1.4), but the

system has disadvantages in homogeneity and decoherence. Donor systems, on the

other hand, are behind in terms of single spin control accomplishments, but have the

promise of increased homogeneity and reduced decoherence, and so are a worthwhile

system to research.


Figure 2.3: Simple diagram of the lambda system used for optically manipulating anelectron spin.

The first optical spin rotations of donor-bound electrons were performed on silicon

donors in GaAs [41]. Unfortunately, researchers were not able to isolate individual

Si donors and the spin rotations suffered from decoherence, both of which were most

likely caused by the small binding energy of the exciton to the neutral donor. Fluorine

donors in zinc selenide have a relatively large exciton binding energy, and so there is

hope that spin rotations will work in this system. Approximately half of my research

time has been spent on the F:ZnSe system, and in Chapter 3, I discuss our results on

the isolation and optical pumping of individual F donors.

Although phosphorus donors in silicon was the first donor system proposed for

quantum information purposes [42], Si is extremely optically inefficient due to it’s

indirect band gap. Therefore, electrical methods for interacting with the donor must

be used, at least in part. Despite this disadvantage, the homogeneity and decoherence

times in this system are unprecedented, and so a great deal of work on this system is

ongoing. The other half of my research time has been spent on the P:Si system, and

so in Chapter 4, I will discuss our research on a novel way to combine optical pumping

and electrical detection in order to detect the nuclear spin of a single P donor.

Chapter 3

Fluorine donors in ZnSe

Single electrons bound to fluorine donors in zinc selenide have many potential advan-

tages over other qubit candidates. As with other donor systems, F-bound electrons in

ZnSe are very homogeneous. It has been recently shown that an ensemble of fluorine

donors in bulk ZnSe features long electron-spin dephasing times T ∗2 , greater than 30 ns

for temperatures up to 40 K [43]. Furthermore, photons emitted from the D0 → D0Xtransition of two different donors are identical [44], and can be entangled [45]. ZnSe

is a direct bandgap semiconductor, and so radiative recombination is the dominant

mechanism for decays between D0X and D0. This optical transition has a quantum

efficiency very close to unity [46], and can function as a source of triggered single pho-

tons [44]. The optical dipole coupling is particularly strong for the F:ZnSe system,

leading to fast optical decay times and strong interaction with applied laser fields.

Zinc and selenium atoms both have isotopes with zero nuclear spin: zinc has 96%

natural abundance and selenium has 94% natural abundance of zero-spin isotopes.

This is a very big advantage over other materials for which there are no zero-spin

isotopes, such as III-V semiconductors, where the nuclear spin of the host semicon-

ductor crystal can couple to the electron spin and cause decoherence. In fact, this is

the leading decoherence-causing mechanism for single electrons in InGaAs quantum

dots [47]. In ZnSe, isotopic purification can be used to deplete nuclear spin from

the ZnSe crystal, and is expected to greatly increase decoherence times. This tech-

nique has been successfully used in both diamond and silicon [48, 40]. Fluorine, on

27

28 CHAPTER 3. FLUORINE DONORS IN ZNSE

the other hand, has 100% natural abundance of spin-1/2 isotopes. The spin of a F

nucleus could therefore be used as a long lived quantum memory which is naturally

coupled to the electron spin.

A further advantage is the ability to implant F donors using ion implantation.

This technique could eventually be used to deterministically place single qubits in

specified locations through the use of implantation through a mask and single-impact

registration [49]. Systems such as self-assembled InGaAs quantum dots have been

difficult to grow in specified locations [50], and would therefore be harder to turn into

a scalable technology. Ion implantation would make the F:ZnSe system much more

scalable. F has been successfully implanted and has been shown to take on the role

of individual donors [51].

All of these properties make flourine-bound electrons in ZnSe strong candidates

for quantum information processing, particularly for quantum repeater technology. In

this chapter, I will explain the optical properties of F-bound electrons, describe how

we were able to isolate single F donors, present experimental results demonstrating the

first optical control of a single donor-bound electron, and discuss future experiments

and the outlook for the system.

3.1 Single donor isolation

In our samples, the F donors were isolated by means of a quantum well and mesa

structuring, as described in Ref. [52] and shown in Fig. 3.1. A ZnSe quantum well

confined between ZnMgSe cladding layers serves to isolate F donors to a 2D plane,

while mesa etching further confines the donors to disk-like quantum well regions.

Based on the areal doping density of the F within the sample and the size of the

mesas, we can control how many donors exist within a particular mesa.

After this, the quantum well, and therefore the F donors with optical transitions

below the ZnSe bandgap, are confined to ∼100 nm diameter mesas, which are sepa-rated by 10 µm.

The samples were grown using molecular-beam epitaxy (MBE) on top of a GaAs

substrate. Unfortunately, good single-crystal ZnSe substrates do not exist, and so

3.1. SINGLE DONOR ISOLATION 29

Figure 3.1: Mesa structure used to isolate individual donors. Donors are δ-dopedwithin the center of the ZnSe quantum well. After growth, the sample is etcheddown through the quantum well to the surface, leaving behind 100 nm-diameter mesastructures.

GaAs was chosen due to the similarity of the lattice constants (5.67 Å for ZnSe [53]

and 5.65 Å for GaAs [54]). On top of that, a layer of ZnSe was grown, usually 20 nm

in thickness, which helps the lattice matching and the adhesion between the GaAs

and the ZnMgSe. The cladding layers of Zn1−xMgxSe generally had up to 17% Mg

and were grown to be a few 10s of nm thick. Between the two ZnMgSe cladding layers

is the ZnSe quantum well with a thickness of 1-10 nm.

The F were placed within the quantum well by either turning on the F source for

a short time during the epitaxial growth, resulting in a δ-doped layer of F, or through

ion implantation following the growth. The epitaxially doped F were located only

within the quantum well, and for many of the samples studied for this thesis, the areal

density was close to 3× 1010 cm−2. The flourine placed using ion implantation had adistribution of depths, and so only a fraction of the F end up within the quantum well.


Fortunately, the only donors with optical transitions within the region of energies we

have been studying are the F located within ZnSe, and any F that end up within

ZnMgSe play no role in our studies. The resulting areal density within the quantum

well was similar to the density of the δ-doped samples. Following the growth and

implantation, the samples were generally annealed in order to reduce the amount

of defects with the sample. Implantation creates many defects, and so annealing is

usually required for those samples.

Structuring was done using electron-beam lithography and wet etching in order

to fabricate the mesa structures. Mesas with 100 nm diameter and a F concentration

of 3×1010 cm−2 within the quantum well result in 2.4 donors on average per mesa.The mesas were separated by 10 µm, ensuring we could isolate optical emission from

individual mesas. After structuring, a layer of silicon dioxide or silicon nitride was

usually deposited on top of the entire sample and acts as a passivation coating. Due to

the lattice mismatch and varying thermal expansion coefficients between the layers in

the sample, samples would often acquire dislocations or fractures after a few thermal

cycles between room temperature and measurement temperatures near 4 K. This

decreases the optical efficiency of emission from the donors. The passivation coating

serves to improve the durability of the samples.

3.2 Optical spectra

The bandgap in ZnSe is wide and has an energy near of 2.8 eV in bulk, which corre-

sponds to photons in the blue with a wavelength around of 440 nm. A single electron

bound to a fluorine donor in ZnSe has a binding energy of 29.3 meV in bulk [55, 56, 57].

This relatively large binding energy means that the electron is usually bound even

up to room temperature. The D0X state has a binding energy of 5 meV in bulk [58],

which means that the donor will only trap an exciton at temperatures significantly

below 60 K. The D0X→ D0 transition in bulk at 4K has an energy of 2.800 eV [58].These emission binding energies are only accurate for bulk ZnSe, and can be notice-

ably blue-shifted when the fluorine donors are located within the quantum well used

in our samples.

3.2. OPTICAL SPECTRA 31

The D0X binding energy can be measured using photoluminescence spectroscopy.

For these experiments, we used an above-band laser (a laser with photon energies

larger than the bandgap of ZnSe) to illuminate the sample. This creates many free

carriers, both electrons and holes, within the sample. Most of these will combine into

excitons and later recombine, emitting a photon. Occasionally an exciton will relax

into the D0X state before recombining, and so it will release a photon with slightly

lower energy, where the difference is the D0X binding energy. The emitted photons

are then collected and sent to a spectrometer. Fig. 3.2 shows an example bulk ZnSe

spectra.

Figure 3.2: Example spectra of a bulk F:ZnSe sample. Here, both the HH and LHfree exciton lines are visible. Slightly lower in energy than the free excitons is thestrong D0X transition peak. Even lower energy is the two-electron-satellite transition(Sec. 2.3), which is due to D0X transition into an excited state of the neutral donor.

When the quantum well thickness approaches the effective bohr radius of the

donor-bound electron, 3.59 nm for F:ZnSe [41] (see Eq. 2.15), the quantum well

starts to compress the electron wavefunction in one dimension, forcing it to spread

out in the plane of the quantum well. This tends to increase the energy of the

D0 state. However, the quantum well also effects the conduction band free-electron


wavefunctions, which begins to look like a particle-in-a-box wavefunction along the

growth direction of the quantum well. This increases the energy of the free-electron

wavefunctions as well. Except in the case of a very narrow quantum well, the increase

of the free-electron energy is greater than that of the D0 energy, resulting in an increase

of the electron binding energy to the F donor [41]. This same effect happens with the

free excitons and donor-bound excitons, resulting in an increase in the D0X binding

energy. The location of the F within the quantum well also has an effect on these

binding energies [41]. See Figs. 3.3 and 3.4 for example 4 nm and 2 nm quantum well

luminescence.

Figure 3.3: Example spectra for a single mesa with a 4 nm thick quantum well. Theentire spectra is blue-shifted compared to the bulk spectra shown in Fig. 3.2. Theset of peaks at higher energy are due to HH free excitons. The large single peak atlower energy is likely due to one or more donors, based upon the energy and widthof the peak.

The quantum well also serves to provide enough strain that the LH D0X states

are split off from the HH D0X states, due to the Pikus & Bir strain Hamiltonian [59].

This puts the HH states higher in the valence band than the LH states, resulting in

a smaller D0X → D0 transition energy. This splitting is large enough that emissionfrom the LH D0X states are never observed from quantum well samples. Therefore,


Figure 3.4: Example spectra for a single mesa with a 2 nm thick quantum well.The spectra is blue-shifted even further in the narrower quantum well. The narrowquantum well also amplifies the relaxation of excitons into the neutral-donor state,increasing the amplitude peak corresponding to that transition.

we only need to worry about the spin-3/2 states, and the full optical system that we

consider in most situations is two D0 states and two D0X states, both defined by their

spin states.

The minimum width of the optical transition is determined by the lifetime of

excited state with the relation

∆ν =1

2πτ(3.1)

where ∆ν is the linewidth in Hz, and τ is the lifetime in s. This lifetime-limited

transition has a Lorentzian shape, since the fourier transform of an exponential decay

is a Lorentzian curve. However, the transition can be broadened by inhomogeneity,

such as the distribution of emission energies from a large ensemble of donors, or by

the time averaged emission of a donor that has a fluctuating transition energy. This

inhomogeneous broadening generally results in a Gaussian shaped curve rather than

a Lorentzian curve.


3.2.1 Magnetophotoluminescence

When placed in a magnetic field, both the D0X and D0 states split due to Zeeman

splitting. The orientation of the magnetic field with respect to the quantum well

growth direction plays a big role in determining the wavefunction of the donor states,

which determine the transition selection rules. There are two orientations we work in.

In the first, the magnetic field is parallel to the growth direction, and in the second,

the magnetic field is perpendicular to the growth direction. Due to the fact that our

optical axis is always parallel to the growth direction, the first orientation is Faraday

geometry, and the second is Voigt geometry.

Using magnetophotoluminescence, we can show that the D0X optical transitions

do in fact follow the selection rules discussed in Sec. 2.3 and computed in Sec. A.1.

The experiments are the same as the photoluminescence experiments, using above-

band laser illumination and a spectrometer for collecting photon emission, with the

addition of an applied magnetic field and polarization-selective measurements. The

polarization sensitivity is created by using a wave-plate in front of a polarizing beam

splitter (PBS) with the H-polarization output of the PBS leading to the spectrometer.

To detect V-polarization photons |V 〉, we use a half-wave plate (HWP) with the fast-axis at an angle of 45◦ to the horizontal, which changes |V 〉 into |H〉 before the PBS.Using the HWP at 0◦, |H〉 remains |H〉, thus measuring H-polarization photons. Wecan also measure in the circular basis |σ+〉/|σ−〉 using a quarter wave plate (QWP)with the fast axis at +45◦/-45◦ with respect to the horizontal.

Using a magnetic field of 7 T, we can observe D0X transitions with the expected

polarization in either Faraday or Voigt geometry, as shown in Fig. 3.5. By fitting

the splitting between the peaks as a function of magnetic field, we can determine

the g-factor of the electron and the hole. Since the hole spin is unable to align with

the magnetic field in Voigt geometry, the magnetic field does not cause any Zeeman

splitting, and so the hole g-factor is approximately zero. By comparing the energy of

four optical transitions in Voigt geometry, the splitting between the two D0 and two

D0X states can be determined. This determines both the electron g-factor, ge, and

the hole g-factor in a magnetic field perpendicular to the growth direction, g⊥h . Since

the electron wavefunction is not confined by the quantum well, the electron g-factor


is the same in both Faraday and Voigt geometry. Therefore, we can compare the

electron g-factor from Voigt geometry to the splitting of the two optical transitions in

Faraday geometry in order to determine the hole g-factor in a magnetic field parallel

to the growth direction, g‖h.

Figure 3.5: Spectral data showing the D0X transition of a single donor at 7 T in bothFaraday and Voigt geometry. The points are the data, and the lines are Gaussiancurve fits. For the Faraday plot, the blue curve is R circular polarization, and the redcurve is L circular polarization. For the Voigt plot, the blue curve is H polarization,and the red curve is V polarization.

The g-factor also depends upon the thickness of the quantum well, in addition to

the orientation. This is because the g-factors for ZnSe are slightly different than those

in ZnMgSe, and so if the electron or hole wavefunction leaks out into the quantum well

barriers, the g-factor will change. In the case of the hole, the spin-orbit coupling is also

changed as the wavefunction is compressed in the quantum well, further modifying

the g-factor.

3.2.2 Single photon source

Since the donor emits one and only one photon for each decay from the D0X state

and since the donor can only trap one exciton, it makes for a great single photon


source. When the D0X state is exciting with an above-band pulsed laser, the donor

is a triggered single-photon source. Each pulse creates many excitons, and one of

them can bind to the donor. This exciton will eventually decay, emitting a single

photon. However, the exciton lifetime is shorter than the D0X lifetime [44], and so

the remaining excitons will decay as free excitons before the D0X emits the photon.

Therefore, there are no remaining excitons to recreate the D0X state after it emits

the first photon, and there will be at most one photon emitted by the donor for each

excitation pulse.

The single-photon nature of the donor system can be confirmed by determining

the two-photon correlation function g2(τ),

g2(τ) =

〈a†(t)a†(t+ τ)a(t+ τ)a(t)

〉〈a†a〉2

. (3.2)

This function can be measured using a Hanbury-Brown-Twiss (HBT) experiment [60],

as depicted in Fig. 3.6.

Figure 3.6: Simple diagram of the Hanbury-Brown-Twiss experiment. A pulsed laser(purple) excites the D0X state within the sample. The emitted photons are collectedand sent to a beam splitter. If there is at most one photon at any time, it is impossibleto detect a photon on each detector at the same time.

In this experiment, the sample is excited by an above-band pulsed laser. The

emission is filtered by frequency so that all of the photons from the pump laser are


blocked, and the only part of the spectrum detected is the narrow frequency range

associated with the D0X transition. We can then imagine a regular train of single

photons emitted from the sample and heading to the beam splitter. When the photons

hit the beamsplitter, they have a 50% probability of being sent to either SPCM. When

the SPCM detects a photon, a timer starts. The timer is stopped when the second

SPCMs detects a photon. A histogram of the time between photon arrivals produces

a plot of g2(τ) (after normalizing the plot by setting g2(τ) = 1 for τ far from zero

delay). Since a single photon cannot be split and detected by both SPCMs at the

same time, a single photon source will always have g2(0) = 0.

Unfortunately, in a real experiment, it is often hard to filter out all the photons

at other energies, such as those that come from other light sources in the room and

especially those from any laser incident on the sample. This will result in the g2(0)

dip not going all the way to zero. The threshold for confirming a single photon source

is for g2(0) to be less than 0.5, since 0.5 would correspond to the case where two

photons were always emitted together. If the measured dip is less than 0.5 by a few

standard deviations, we can be quite confident that the emitter is a single photon

source. Fig. 3.7 shows a measurement of g2(τ) using the experiment described above,

resulting in g2(0) = 0.25. From this result, we know that the photoluminescence

Figure 3.7: Example g2(τ) data for a single donor. At τ = 0, the normalized dipdrops to 0.25. This figure is taken from Ref. [52].


Figure 3.8: Simple diagram of the Hong-Ou-Mandel experiment. In this case, twodifferent donors are excited by the pulsed laser. The emitted photons are collectedand sent to the same beam splitter. If identical photons arrive at the same time, theywill bunch together and travel to the same detector.

emission we were detecting was coming from a single photon source.

A slight variation of the above experiment can be used to show that the emitted

photons are indistinguishable. This is called the Hong-Ou-Mandel (HOM) experi-

ment [61], and is depicted in Fig. 3.8. When two perfectly indistinguishable photons

hit a beam splitter at the same time from opposite sides, they will bunch together

and always exit the beam splitter on the same side.

a†b† |0〉 = 12

(c† + d†)(c† − d†) |0〉 = c†c† − d†d†

2|0〉 . (3.3)

Therefore, at zero time delay, g2(0) should go to zero for indistinguishable photons.

However, in reality the photons will not always arrive at the same time even if

they are detected at the same time due to the finite lifetime of the excited state

3.3. SINGLE DONOR CONFIRMATION 39

causing jitter in the photon arrival time and the detector jitter reducing the preci-

sion of the measurement of the arrival time. In addition, the center frequency or

wavepacket shape can vary slightly. These effects result in a reduced indistinguisha-

bility. In experiments, photons from two different fluorine donors have shown good

indistinguishability [44]. As shown in Sec. 1.3.1, the interference of indistinguishable

photons can be used to entangle qubits separated over macroscopic distances. The

first step of this scheme, which is the post-selected entanglement of photons from a

pair of donors, has also recently been demonstrated [45].

3.3 Single donor confirmation

For the optical pumping experiments described in the next couple of sections, a

mesa containing a single F donor needed to be found. To confirm the presence of a

single isolated F donor, we used four experiments on the same mesa. T

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